First off, research is important. You need to determine the ownership of the cemetery in order to seek permission. Your best bet probably would be to first, contact the owner to gain access to the cemetery to do further research and analysis, if you have not already done so.

With a plan in hand, seek permission from the owner to begin work. I think you're better off having a plan when seeking this permission, as opposed to just asking without any discernable idea of what you actually want to do. The IHPA has a Permission to Clean and Preserve a Historic Illinois Cemetery form that gets this agreement in writing and starts the permit-obtaining process (basic work, including cleaning or repairing markers requires a permit in Illinois). Also note "that you are required to demonstrate that you have received proper training in basic cemetery preservation methods before you will be authorized to continue. Work can not begin until you have received a written permit from the IHPA." (IHPA FAQs)

The IHPA has an FAQ page that answers questions specific to Illinois cemeteries, but also discusses, in general, why cemeteries become abandoned and why preservation is important. This is also a great read!